-- produce a nullhomotopy for a map f of chain complexes.
Whether f is null homotopic is not checked.
Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.
i1 : A = ZZ/101[x,y];
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i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | 16x2-15xy-26y2 -33x2-24xy+44y2 |
| -15x2+40xy+26y2 -18x2+20xy+8y2 |
| -3x2+9xy+31y2 -25x2+18xy+12y2 |
3
o2 : A-module, quotient of A
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i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
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i4 : N = prune (M**R)
o4 = cokernel | 34x2+21y2 -7x2+5xy-29y2 x3 x2y+37xy2+8y3 33xy2-3y3 y4 0 0 |
| x2-41xy-46y2 -44xy-48y2 0 41xy2 -48xy2+24y3 0 y4 0 |
| 41xy-13y2 x2-6xy+21y2 0 -5y3 xy2+32y3 0 0 y4 |
3
o4 : A-module, quotient of A
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i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
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i6 : d = C.dd
3 8
o6 = 0 : A <------------------------------------------------------------------------ A : 1
| 34x2+21y2 -7x2+5xy-29y2 x3 x2y+37xy2+8y3 33xy2-3y3 y4 0 0 |
| x2-41xy-46y2 -44xy-48y2 0 41xy2 -48xy2+24y3 0 y4 0 |
| 41xy-13y2 x2-6xy+21y2 0 -5y3 xy2+32y3 0 0 y4 |
8 5
1 : A <----------------------------------------------------------------------- A : 2
{2} | 8xy2 -48xy2-13y3 -8y3 -17y3 -47y3 |
{2} | -31xy2-43y3 -12y3 31y3 12y3 -43y3 |
{3} | -9xy-6y2 21xy+30y2 9y2 -11y2 -17y2 |
{3} | 9x2-21xy-24y2 -21x2-45xy+7y2 -9xy+27y2 11xy+45y2 17xy-44y2 |
{3} | 31x2-4xy+15y2 -44xy+10y2 -31xy+47y2 -12xy-4y2 43xy-27y2 |
{4} | 0 0 x-18y 30y 11y |
{4} | 0 0 -8y x-9y -43y |
{4} | 0 0 -3y -19y x+27y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
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i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------ A : 0
{2} | 0 x+41y 44y |
{2} | 0 -41y x+6y |
{3} | 1 -34 7 |
{3} | 0 36 -45 |
{3} | 0 5 29 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <--------------------------------------------------------------------------- A : 1
{5} | -36 -27 0 8y -13x-4y xy-4y2 46xy+10y2 -47xy-8y2 |
{5} | 44 44 0 24x-48y -20x-18y -41y2 xy-28y2 48xy+45y2 |
{5} | 0 0 0 0 0 x2+18xy-50y2 -30xy-9y2 -11xy+21y2 |
{5} | 0 0 0 0 0 8xy+42y2 x2+9xy-49y2 43xy+47y2 |
{5} | 0 0 0 0 0 3xy+24y2 19xy-28y2 x2-27xy-2y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
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i8 : s*d + d*s
3 3
o8 = 0 : A <---------------- A : 0
| x3 0 0 |
| 0 x3 0 |
| 0 0 x3 |
8 8
1 : A <----------------------------------- A : 1
{2} | x3 0 0 0 0 0 0 0 |
{2} | 0 x3 0 0 0 0 0 0 |
{3} | 0 0 x3 0 0 0 0 0 |
{3} | 0 0 0 x3 0 0 0 0 |
{3} | 0 0 0 0 x3 0 0 0 |
{4} | 0 0 0 0 0 x3 0 0 |
{4} | 0 0 0 0 0 0 x3 0 |
{4} | 0 0 0 0 0 0 0 x3 |
5 5
2 : A <-------------------------- A : 2
{5} | x3 0 0 0 0 |
{5} | 0 x3 0 0 0 |
{5} | 0 0 x3 0 0 |
{5} | 0 0 0 x3 0 |
{5} | 0 0 0 0 x3 |
3 : 0 <----- 0 : 3
0
o8 : ChainComplexMap
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i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|